-
6 (200
twati
an
East
Univ
ised f
brid
nationwide bridge replacements and rehabilitations are
for structural safety and serviceability for an existing
ing into account the live load increase and material
rating of the bridge. Rating factors for dierent dete-
ces for dierent deteriorating bridgthe same failure m
oad model and deterg author. Tel.: +1-303-4steel are
integratess: dan.frangopol@colorad Correspondin7317.
E-mail addre0141-0296/$ - see front matter# 2004
Elsedoi:10.1016/j.engstruct.2004.06.012calculated forpose, a live
lconcrete and
92-7165; fax: +1-303-492-
o.edu (D.M. Frangopol).vier Ltd. All rights reserved.stem
reliability indi-e types must also beodes. For this pur-ioration
models ford into a speciallymembers, or addition failure modesment
work, change in strength ofriorating bridge types have to be
calculated for various
. Consequently, the sybridge is determined when a maintenance,
improve-based on ratings listed in the National Bridge Inven-tory
(NBI) database. For distribution of funds, High-way Bridge Repair
and Replacement Program(HBRRP) uses the so-called suciency rating,
which iscalculated by a formula incorporating structural
safety(55%), serviceability and functional obsolescence(30%), and
essentiality for public use (15%) [14]. Pos-sessing the highest
weight in suciency rating formula,load rating for structural safety
is a crucial measure forbridge management and decision making. Load
rating
deterioration models, are more commonly used for life-time
bridge assessment.For the design of new bridges, the AASHTO
LRFD
[1] Specications provides the necessary provisions forobtaining
a uniform safety level for bridge compo-nents. Once a bridge is
designed and placed in service,the AASHTO Manual for Condition
Evaluation ofBridges [2] provides provisions for determination of
thesafety and serviceability of existing bridge components.The
minimum of the component ratings determines thein the National
Bridge Inventory database. Distribution of funds is based on the
suciency rating, represented by a formula con-sidering structural
safety, functional obsolescence, and essentiality for public use.
Possessing the highest weight in suciency rat-ing formula, load
rating is a crucial measure for bridge management. While load
rating represents the current practice in bridgeevaluation,
reliability methods, taking into account live load increase and
material deterioration models, are more commonly usedfor lifetime
bridge assessment. In this paper, time-dependent relationship
between the reliability-based analysis results, represent-ing the
future trend in bridge evaluation, and the load ratings is
investigated for dierent types of bridges located within an
exist-ing bridge network. The comparisons between live load rating
factors and reliability indices are made over the lifetime of
eachbridge in the network. The ratingreliability prole and
ratingreliability interaction envelope concepts are introduced.
Further-more, the ratingreliability proles are collectively
examined in order to evaluate the time-dependent performance of the
overallbridge network.# 2004 Elsevier Ltd. All rights reserved.
Keywords: Bridges; Bridge network; Deterioration; Rating;
Reliability; Structures; System reliability; Time-dependent
performance
1. Introduction
Prioritization and allocation of federal funds for
of dead load alters the condition or capacity of thestructure.
While load rating represents the currentpractice in bridge
evaluation, reliability methods, tak-Engineering Structures 2
Time-dependent interaction beof deterior
Ferhat Akgul a, Da Department of Engineering Sciences,
Middle
b Department of Civil, Environmental, and Architectural
Engineering,
Received 8 December 2003; received in rev
Abstract
Prioritization and allocation of federal funds for nationwide4)
17511765www.elsevier.com/locate/engstruct
een load rating and reliabilityng bridges
M. Frangopol b,
Technical University, 06531 Ankara, Turkey
ersity of Colorado, Campus Box 428, Boulder, CO 80309-0428,
USA
orm 12 June 2004; accepted 18 June 2004
ge replacements and rehabilitations are based on ratings
listed
-
portation (CDOT) consists of a letter followed by a
of each bridge in the network. This is followed bydetermination
of initial component and system
1752 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765Fig. 1. Bridge network.reliability indices based on system
failure models.Detailed descriptions of these procedures are
presentedin Akgul and Frangopol [4,6]. Finally, in order to
cal-culate the rating factors and reliability indices forbridge
components and systems over time, time-depen-dent live load model
and corrosion models for concreteand steel are adopted. The
theoretical bases for thesemodels are given below. In order to
explain how theresults are obtained for a typical bridge, as a
represen-tative example, the results of the live load model,
cor-rosion deterioration model, and time variation of
thereliability index and the rating factor for the compo-nents of
bridge E-17-LE are also presented in the fol-lowing sections.
3.1. Time-dependent live load model
Live load on a bridge is a function of many para-meters, such as
truck weight, axle loads, axle congur-ation, span length, position
of vehicles (longitudinal,one to two digit number describing the
vertical andhorizontal coordinates of the bridge, respectively, on
astatewide grid. The last two letters are the uniqueidentication
symbol for the bridge [10].Characteristics of the network bridges
are listed in
Table 1 (see also [4]). The bridge network consists ofseven
prestressed girder, three steel rolled I-beam, andfour combined
welded steel plate and reinforced con-crete girder bridges. Table 1
lists data such as thelength and width of each bridge in addition
to the yearin which it was built. The lengths range between 34.1and
82.3 m. The oldest and newest bridges in the net-work were built in
1951 and 1995, respectively, repre-senting a 44-year span between
their constructions.
3. Bridge live load, deterioration, and system
reliability
Time-variation of the rating factor and the reliabilityindex are
determined based on the fact that while thelive load is increasing
due to larger number of truckspassing on the bridge, cross
sectional areas of steelreinforcement and structural steel in
bridge decks andcomponents will be gradually decreasing starting
fromthe onset of corrosion on their surface. A step-by-stepprocess
is used to calculate the time-variation of therating factor and the
reliability indices. First, initialrating factors are calculated
for each bridge componentbridges located within close proximity of
each other,preferably along interstate highways, and within thesame
transportation and maintenance regions. Bridgedesignation used by
Colorado Department of Trans-developed computer program [5]. This
program is usedto determine the lifetime reliability proles, while
life-time rating calculations are performed
separately.Time-dependent relationship between the reliability-
based analysis results, representing the future trend inbridge
evaluation, and the load ratings, reecting thecurrent practice in
bridge evaluation, is investigatedherein for existing bridges
located within a bridge net-work. The comparison between live load
rating factorsand reliability indices are made over the lifetime
ofeach bridge in the network. The so-called ratingreliability prole
and ratingreliability interactionenvelope concepts are introduced.
Furthermore, theratingreliability proles are collectively examined
inorder to evaluate the time-dependent performance ofthe overall
bridge network.
2. Bridge network
The bridge network shown in Fig. 1 [3] is located atnorthwest
corner of Denver metropolitan area, andconsists of 14 mixed type
highway bridges. This net-work was described in Akgul and Frangopol
[4,6]. Theselection criteria for the bridge network was to
choose
-
bridges. Other primary causes for bridge deterioration
in Fig
ength
)
.1
.9
.5
.1
.3
.6
.7
.2
.0
.3
.6
.5
.0
.7
uous
ompo
F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765 1753transverse), number of vehicles, speed of vehicles,
sti-
ness of superstructure, and bridge geometry (straight,
curved) [16]. Site specic trac data for such para-
meters is generally collected using weigh-in-motion
(WIM) studies. WIM involves recording weights of
trucks and bridge deections by means of sensors
attached to bridge deck and girders. The results are
used to quantify the load and resistance values such as
actual load eects and girder distribution factors for
the bridge. Subsequently, actual recorded eects are
generally compared to values specied by current
bridge design codes.In this study, the load model developed for
the
AASHTO LRFD Bridge Design Specications [1] is
Table 1
Characteristics of 14 Colorado highway bridges in the network
shown
Bridge
name
Bridge
type
CDOT
designation
Number
of spans
L
(m
E-16-MU Prestressed CPG 1 34
E-16-LA Prestressed CBGCP 2 77
D-16-DM Prestressed CPGC 2 44
E-16-QI Prestressed CBGCP 2 74
E-16-LY Prestressed CPGC 3 74
E-16-NM Prestressed CPGC 2 64
E-17-MW Prestressed CIC 2 72
E-16-FK Steel I-beam CIC 4 69
E-16-FL Steel I-beam CIC 4 54
E-16-Q Steel I-beam CIC 5 82
E-17-LE Steel plate girder WGCK 4 68
E-17-HS Steel plate girder WGCK 4 64
E-17-HR Steel plate girder WGCK 4 64
E-17-HE Steel plate girder WGCK 4 67
CPG, concrete prestressed girder; CBGCP, concrete box girder
contin
crete on rolled I-beam continuous; WGCK, welded girder
continuous cused. The extrapolation for the load model used
herein
standard AASHTO truck load. A drastic increment inmay be the
cracking, dislocation at supports, bearing
damage, excessive vibration, delamination of decks,
and heavy truck use. This study focuses on deterio-the rst few
years is followed by a gradual increase
over the lifetime of the bridge.
3.2. Deterioration models for concrete and steel
Two essential materials used for bridge construction,
concrete and steel, are both aected by environmental
stressors. They deteriorate progressively when exposed
to atmosphere and chlorides. Deterioration of these
materials is one of the major causes of deterioration of
. 1 (see also [4])
Width Year built ADTT
(trucks/day)(ft) (m) (ft)
112.0 11.6 38.0 1994 810
255.5 39.2 128.5 1983 450
146.0 14.2 46.5 1990 390
243.2 30.7 100.7 1995 1335
243.7 34.1 112.0 1985 1610
212.0 28.0 92.0 1991 2955
238.6 30.5 100.0 1987 230
227.0 10.4 34.0 1951 1370
177.0 10.4 34.0 1951 765
270.0 12.2 40.0 1953 890
225.0 19.7 64.5 1972 992
211.7 10.4 34.0 1963 5
209.8 10.4 34.0 1962 306
222.2 10.4 34.0 1962 1290
prestressed; CPGC, concrete prestressed girder continuous; CIC,
con-
site.ration due to corrosion only, since time variant models
d relative humidity),
sure nitia
orientation, angle of exposure, time of wetness, atmos-the
multiplier (uncertainty factor) for exural moment
in slab is shown for the 75-year time period for the pheric
pollutants, deicing salt, and debris) [7,12]. ForFig. 2 demo
slab of bridgenstrates how this
E-17-LE. In thismodel is used
gure, the varfor the
iation of and expoconcrete deteconditions (irioration model, Fl
climate, sicks secoheltering,corrosion include temperature an
initial time t 0, i.e. due to a single truck. atmosphere
(environmental conditions aecting steel
standard deviation of maximum moment or shear at on the metal
(composition of alloys in metal), local
ber (0.577216), and lX and rX are the mean value and action.
Severity of steel corrosion, in general, depends
of Type I largest value distribution, c is the Euler num-
where un and an are the location and scale
parametersalkali-silica reaction, sulphate attack and
freezethawfrequently observed, followed by deterioration due to6an
by carbonation and chloride contamination are most
rYn p rX 2 Refs. [8,9,13,15]. For steel corrosion, the types
causedpYn X n X anXmodels for steel corrosion in concrete can be
found in
l l u r c r 1 this eld. For instance, treatment of various types
ofvarious models have been developed by researchers intime,
corresponding to a sample size of n trucks:moments and shears in
bridge components at a future For deterioration of concrete and
steel in bridges,is based on the mean value lYn and standard
deviationrYn of Type I largest value asymptotic distribution
for
for other deterioration types have not yet been fully
developed in this eld.nd law of
-
Fig. 2. Time-variation of the mean of the maximum moment in the
slab of bridge E-17-LE.
1754 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765diusion due to chloride is used:
@C
@t Dc @
2C
@x23
where C is the concentration of chloride ions, Dc is theeective
diusion coecient, t is time, and x is the dis-
tance from outer surface of the solid.
Fig. 3. Time-variation of the slab reinModels developed to
predict time-dependent cor-
rosion penetration in steel are usually empirical for-
mulas intending to capture the actual corrosion
process. They generally include the time variable
accompanied by several regression coecients in the
form of a power formula. In this study, a power func-forcing
area for bridge E-17-LE.
-
tion is em
p botb1where boyears, resmic transThe d
second lacrete deckconcretesteel (4) ithe applicthe slabdom
vartime foryears. Withe corrorcorr indic
ated forstructureders) areple, theridge E-for the
and (b),
e modele bridgean pre-is con-
s, exureons arebridges,ed bothre of thethe slab
tem f
F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765 1755and employs the First Order System ReliabilityMethod
to determine the reliability proles for bridgemembers and systems.
System reliability analysis of thebridges required the description
of a failure model for
Fig. 4. (a) Girder layout, (b) systributions of the slab
reinforcement area are plotted at10-year intervals using the Monte
Carlo simulationmethod. The prole shown represents the mean valueof
these distributions over time.
3.3. System reliability model
Once the models for live load and material deterio-ration (for
dierent bridge member types) were ident-ied, they were implemented
as program modules intothe network-level lifetime system
reliability programRELNET (reliability of system networks) [5].The
program uses the Monte Carlo simulation tech-
nique to calculate the resistance degradation proles
the network. As representative examples, time variationof the
reliability index and the operating rating factorfor the slab and
girder of bridge E-17-LE are presentedin Figs. 5 and 6. Once these
proles are obtained foreach bridge in the network, the results of
time-dependent rating calculations and reliability proles are
ailure model for bridge E-17-LE.or exural or shear failure of
two adjacent girders.When calculating the system reliability index
for thebridge superstructure, a correlation coecient of 0.5
isassumed between girder resistances, representing themidpoint
between the ideal cases of independence andperfect correlation.
4. Time-dependent rating factor vs. reliability index
for individual bridges
The proles for resistance degradation and reliabilityindex are
systematically generated for each bridge inployed for corrosion of
structural steel [11,17]
4and p are the corrosion losses after one and tpectively, and b1
is the slope of the logarith-formation of (4).eterioration model
associated with Ficksw of diusion (3) is used for reinforced con-s,
reinforced concrete girders, and prestressedgirders. Similarly, the
corrosion model fors applied to steel girders. Fig. 3
demonstratesation of the concrete deterioration model
forreinforcement for bridge E-17-LE. The ran-iable Tisr, indicating
the corrosion initiationslab reinforcement has a mean value of
3.61th the lognormal (LN) density distributions ofsion initiation
time Tisr and corrosion rateated in the gure, the probability
density dis-
each bridge type. Since bridges are generally rsuperstructure
components, only the supermembers of the bridges (i.e., slab and
the girconsidered in this investigation. As an examcross sectional
view of the superstructure of b17-LE and the corresponding failure
modelsuperstructure are displayed in Fig. 4(a)respectively.The
topology of series-parallel system failur
for the superstructure changed depending on thtype. For
instance, for girders of simple spstressed bridges, exure of girder
at midspansidered while for continuous prestressed girderat both
midspan and pier support locatiincluded in the system failure
model. For steelsuch as E-17-LE, girder failure modes includexure
and shear. As shown in Fig. 4(b), failusteel bridge is dened as the
exural failure of
-
Fig. 5. Time-variation of the reliability index for the slab and
girder of bridge E-17-LE.
1756 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765combined, emphasizing the interaction between rating
and reliability proles.Time variation of the operating rating
factor RF(t) vs.
system reliability index b(t) for several representative
bridges of dierent types in the network are presented inFig. 6.
Time-variation of the operating rating factFigs. 713. Network
bridges include a variety of girder
types such as the prestressed concrete, reinforced con-
crete, steel rolled I-beam, and steel welded plate girder
bridges. Girder type is indicated in each graph. Bridge
operating rating factor is plotted against the systemor for the
slab and girder of bridge E-17-LE.
-
F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765 1757reliability index since the operating rating and
the
reliability index are representative of the overload con-
dition and the safety of a bridge, respectively. Operating
rating factor for a bridge indicates the maximum per-
missible live load value that the bridge can carry. HavingFig.
8. Time-dependent operating rating vsa large number of vehicles at
operating rating weight tra-
veling over the bridge may shorten the life of the
bridge.Time-dependent rating factor vs. reliability index
graphs have inverted horizontal axis for the reliability
index values. Thus, the upper-left hand region in theseFig. 7.
Time-dependent operating rating vs. reliability index for bridge
E-16-MU.. reliability index for bridge D-16-DM.
-
Fig. 9. Time-dependent operating rating vs. reliability index
for bridge E-17-MW.
1758 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765graphs represents the highest rating and reliability,while
lower-right hand region represents the lowest rat-ing and
reliability for a bridge component or system.Therefore, over time,
the point representing bridge
rating factorreliability index pair is expected to move
Fig. 10. Time-dependent operating rating vfrom upper left to
lower right hand corner in all graphsdue to member deterioration
and increase in loadeects due to trac.Since the theoretical basis
used in these graphs is thesame, only the graph for bridge E-16-MU,
shown ins. reliability index for bridge E-16-FK.
-
Fig. 11. Time-dependent operating rating vs. reliability index
for bridge E-17-LE.
F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765 1759Fig. 7, will be explained herein. This will also serve
as a
description of the format used for the other graphs.
For the prestressed concrete bridge E-16-MU, Fig. 7Fig. 12.
Time-dependent operating rating vs. reliability inshows three
plots: slab exure, girder exure and the
system ratingreliability interaction curve denoted by
the label (RF(t) , b(t) ). This designation indicatesbridge
sysdex for slab and concrete girders of bridge E-17-HS.
-
bility
1760 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765that the time-dependent rating factor RF(t)bridge is forthe
bridge, representing the minimum of all superstruc-ture component
ratings, and the time-dependent
Fig. 13. Time-dependent operating rating vs. reliareliability
index b(t)sys is for the system representing theseries-parallel
system failure model for the bridge
ingreliability pand system ratigure are overlvalues for the bt
0, are not splayed considerasingle average tringreliability cuto
immediate apthe graphs startrst month, reexposure for theSince the
sud
values due to inded, the remainles shown in tdeterioration, alto
trac is stillmonth, the pointhe bridge E-16nent ratingreliatime
domain hidseparate points in time are displayed consistently for
allbridges: 1, 6 month, 1, 10, 20, 30, 40, 50, 60, and 75years. For
bridge E-16-MU, these points are clearlyvisible in both the slab
and the girder proles. How-
index for slab and steel girders of bridge E-17-HS.ever, for
several other bridges, it was necessary to dis-play a few
representative points instead of all 10 points
liability graphsrvations can beehavior, there-nt bridge typesm a
broad per-liability curvesroup type. Forthe prestressedE-16-MU
and
y, are generallyability domain.irders over pierity values
areE-17-MW (seetressed concretegreliability fory the extensionexure
toward
-17-LE, shownery low ratingich is indicatedince the slab exure
controls the rat-role for the bridge, the slab exurengreliability
interaction curves in thisapped. The rating vs. reliability
indexridge components at initial time, i.e.hown in these graphs
since they dis-bly higher reliability indices due to auck and
revealed sudden drops in rat-rves at initial stage of service life
dueplication of the trac load [4]. Instead,from a time, chosen as
the end of thepresenting a reasonable trac loadbridge.den initial
drop in ratingreliabilityitial application of trac load is
exclu-ing reduction in ratingreliability pro-hese gures is mainly
due to materialthough increase in live load eect duea contributing
factor. Starting from 1ts in time until 75 year life period for-MU
are indicated along each compo-bility prole. In order to reveal
theden in these graphs, the following 10
mentioned above.By comparing the lifetime ratingre
for dierent bridges, the following obsemade: each bridge
displayed a unique bfore, distinct generalizations for dierewould
not be appropriate. However, frospective, general location of
ratingreshow similarities within a given bridge ginstance,
ratingreliability curves forconcrete bridges, as shown for
bridgesD-16-DM in Figs. 7 and 8, respectivellocated near the middle
of ratingreliFor exure of continuous prestressed glocation, higher
rating and reliabilobserved as shown in Fig. 8. BridgeFig. 9),
however, although being a presbridge, displays a relatively low
ratinthe exure of slab which is indicated bof the ratingreliability
curve for slabthe lower-right corner of the graph.Steel bridges
such as E-16-FK and E
in Figs. 10 and 11, respectively, have vreliability values for
exure of slabs whsuperstructure. Sby the extension of the
ratingreliability prole toward
-
the lower-right c
period, slab rat
reach the lowe
bridges. This is d
the oldest ones
quently, they
strength reinforcBridges with s
E-17-HS shown
vior similar to th
arate graphs are
concrete and steIn general,
reliability prole
nent having th
reliability. For r
since the bridge
minimum com
reliability point
ingreliability p
reliability analy
used.Bridge rating
determined considering both the slab and girder rat-
ings. Consequently, following the same process, the
nteraction
he network arependent bridgestem reliability17, The time-he
interaction
ting factors forly, is shown ins are listed inperating bridge,
and for 1, 20,gure enablesctor values fortheir lifetimes.stressed
girdertween 1 monththe other hand,ges is reectedg factors for a
liability indicesfor all network bridges, based on girders only,
is shownin Fig. 15, and the corresponding values are listed in
erati
F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765 1761system reliability indices in these graphs were
calcu-
lated using the system failure model including both
slab and girders.
Fig. 14. Time-variation of the bridge opTable 3. Using this
gure, similar observations andcomparisons can be made for the
network bridgesbased on system reliability indices. The gure
displays
ng rating factor for the network bridges.orner of the graph.
Within 1040-year
ingreliability proles of these bridges
st observed value for the network
ue to the fact that the steel bridges are
among the network bridges. Conse-
were built with slabs having lower
ing steel.teel and concrete girders such as bridge
in Figs. 12 and 13, displayed a beha-
at of the steel bridges. For clarity, sep-
provided in these gures for reinforced
el girders of the bridge E-17-HS.the position of the system
rating
was close to the prole of the compo-
e minimum rating and the lowest
ating, this is a reasonable performance
rating in practice is controlled by the
ponent rating. However, from the
of view, the position of the system rat-
role reected the result of the system
sis based on the system failure model
factors in ratingreliability graphs were
5. Time-dependent ratingreliability i
at network-level
In Figs. 14 and 15, all bridges in tplotted together, showing
the time-deoperating rating factors and the syindices,
respectively. In Figs. 16 anddependent network results show
tbetween rating and reliability proles.The variation of operating
bridge ra
all network bridges, based on girders onFig. 14, and the
corresponding valueTable 2. In Fig. 14, the variation of orating
factors are indicated for 1 month50, and 75 years for each bridge.
Thiseasy comparison of the bridge rating faall bridges in the
network throughoutLarge lifetime deteriorations for prebridges are
characterized by big gaps beand 75 year marks for each bridge.
Onsmall deterioration for steel girder bridby very small vertical
gaps among ratingiven bridge.Similarly, the variation of system
re
-
Fig. 15. Time-variation of the system reliability index for the
network bridges.
1762 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765the snapshots of system reliability indices for allbridges
at various discrete points in time.Figs. 16 and 17, on the other
hand, provide a sum-mary of the ndings reported in Figs. 14 and 15
in
Fig. 16. Time-variation of the bridge operating rating factor
vs. systemterms of bridge operating rating factor vs.
systemreliability index for all bridges in the network consider-ing
slab and girders, and girders only, respectively. In
Figs. 16 and 17, rating reliability envelopes are
shownreliability index for the network bridges based on slab and
girders.
-
Fig. 17. Time-variation of the bridge operating rating factor
vs. system reliability index for the network bridges based on
girders only.
F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765 1763for the network bridges both for time t 1 month andfor
the lifetime interaction proles. The envelope for
time t 1 month, represented by the square region, isa snapshot
of ratingreliability values for all bridges inthe network at that
initial time. The remaining data
Table 3
Time-variation of th
Time b(t)sys
E-16-
MU
1 month 3.73
1 year 3.48
20 year 3.22
50 year 2.67
75 year 1.81
Table 2
Time-variation of th
Time RF(t)OPE,
E-16-
MU
1 month 1.565
1 year 1.442
20 year 1.328
50 year 1.202
75 year 0.983points represent the ratingreliability values of
the
bridges at subsequent discrete points in time through-
out the lifetime of each bridge. Therefore, the network-
level time-dependent ratingreliability interactionenvelope,
represented by upper and lower bounds,
E-17-
HR
E-17-
HE
3.00 3.46
2.70 3.20
2.07 2.89
0.89 2.03
0.08 0.88
E-17-
HR
E-17-
HE
0.802 0.835
0.736 0.773
0.673 0.711
0.507 0.675
0.377 0.479e system reliability index for the network
bridges
E-16-
LA
D-16-
DM
E-16-
QI
E-16-
LY
E-16-
NM
E-17-
MW
E-16-
FK
E-16-
FL
E-16-
Q
E-17-
LE
E-17-
HS
3.28 4.04 4.07 2.59 3.92 3.40 3.33 2.67 2.66 3.47 3.94
3.06 3.75 3.84 2.34 3.67 3.13 3.04 2.36 2.35 3.15 3.54
2.82 3.18 3.59 2.08 3.40 2.85 2.74 2.03 2.02 2.79 2.76
2.24 2.06 2.87 1.95 2.43 2.74 2.65 1.93 1.92 2.68 1.54
1.32 1.35 1.81 1.48 1.34 2.45 2.60 1.89 1.87 2.62 0.56
e bridge operating rating factor for the network bridges
bridge
E-16-
LA
D-16-
DM
E-16-
QI
E-16-
LY
E-16-
NM
E-17-
MW
E-16-
FK
E-16-
FL
E-16-
Q
E-17-
LE
E-17-
HS
1.495 1.595 1.793 1.143 1.616 1.469 0.870 0.763 0.762 0.939
0.949
1.370 1.467 1.656 1.060 1.500 1.339 0.807 0.706 0.705 0.865
0.883
1.256 1.276 1.529 0.982 1.389 1.225 0.746 0.651 0.649 0.790
0.754
1.097 0.953 1.355 0.952 1.184 1.190 0.729 0.635 0.633 0.767
0.563
0.828 0.786 1.063 0.877 0.895 1.133 0.721 0.627 0.626 0.756
0.433
-
(1) Lifetime rating and reliability analyses for dier-
Consequently, based on the time-dependent rating
1 to 0.75, and 93% for reliability, i.e. from 4.1 to 0.25).
1764 F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765ent bridge types in an existing network have to
incor-porate time-dependent models for both live loadincrease and
resistance deterioration. A live load modeland separate
deterioration models for concrete and steeldue to environmental
stressors, such as chloride ingress,for dierent member types were
investigated and theintegration of these models into a
network-level, time-dependent, system reliability analysis program
isaccomplished. When updated with eld data, such pro-grams can
become highly crucial tools for lifetimebridge evaluations in
future practice, and can be furtherimproved to aid maintenance and
repair decisions.(2) Currently, bridge ratings are calculated at
dis-
crete and irregular time periods determined by changesin loading
and/or capacity of the structure. Althoughrecent load and
resistance factor rating procedures aredeveloped with the aim of
incorporating time-depen-dent changes in structural loads and
resistance in theform of load and resistance factors, live load
increaseand structural deterioration are not directly taken
intoconsideration during the rating process (i.e., usingsound and
proven analytical and experimental models).This study demonstrated
that it is possible to predictthe load rating and reliability index
of a bridge usinglive load and resistance deterioration models
integratedinto a single computational platform.shows the safety of
the network bridges throughouttheir lifetime.Fig. 16 revealed an
extremely close behavior for all
bridges considering the lifetime system ratingreliabilityproles.
Although having unique initial ratingreliability values slightly
dispersed at the beginning,ratingreliability values for the network
bridges, basedon both the slab and the girders, followed almost
thesame path when member degradation and live loadincrease were
taken into account. A lifetime ratingreliability interaction
envelope is also dened for thenetwork bridges as indicated in Fig.
16. Based on thisgraph, it is possible to conclude that the
networkbridges show variability in system ratingreliabilityvalues
initially, i.e. t 1 month, however, after the rstmonth of trac, the
values for all bridges on the aver-age gradually converge.A similar
behavior is also observed in Fig. 17, where
the ratingreliability values are based on girders only.In this
case, however, there is a visible distinction,characterized by a
wide gap between lifetime ratingreliability proles of prestressed
and steel bridges. Pre-stressed bridges converged toward rating and
reliabilityindex values signicantly larger than those
associatedwith steel bridges.
6. ConclusionsSince system reliability index evaluates the
actual safety
of the structure, while rating factor reects the live load
capacity only, such a drastic reduction in actual safety
of the structure deserves more attention as compared to
the reduction in live load capacity. Therefore, it may be
more appropriate to base lifetime bridge evaluation on
reliability index rather than the load rating.
Acknowledgments
The partial nancial support of the US National
Science Foundation through grants CMS-9912525 and
CMS-0217290 is gratefully acknowledged. The support
provided by the Colorado Department of Transpor-
tation is also gratefully acknowledged. The opinions
and conclusions presented in this paper are those of the
writers and do not necessarily reect the views of the
sponsoring agencies.
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F. Akgul, D.M. Frangopol / Engineering Structures 26 (2004)
17511765 1765
Time-dependent interaction between load rating and reliability
of deteriorating bridgesIntroductionBridge networkBridge live load,
deterioration, and system reliabilityTime-dependent live load
modelDeterioration models for concrete and steelSystem reliability
model
Time-dependent rating factor vs. reliability index for
individual bridgesTime-dependent rating-reliability interaction at
network-levelConclusionsAcknowledgmentsReferences